U.S. patent number 4,562,426 [Application Number 06/439,740] was granted by the patent office on 1985-12-31 for symbol coding apparatus.
This patent grant is currently assigned to Codex Corporation. Invention is credited to George D. Forney, Jr..
United States Patent |
4,562,426 |
Forney, Jr. |
December 31, 1985 |
**Please see images for:
( Certificate of Correction ) ** |
Symbol coding apparatus
Abstract
In apparatus for transmitting digital symbols over a
band-limited channel using a modulated carrier system by encoding
the symbols into discrete signals selected by a finite state
encoder from a signal alphabet comprised of subsets each
corresponding to transitions from a particular previous state to a
particular current state of the encoder, the improvement in which
at least two of the subsets have a signal in common, and at least
one signal belongs to only one of the two subsets.
Inventors: |
Forney, Jr.; George D.
(Cambridge, MA) |
Assignee: |
Codex Corporation (Mansfield,
MA)
|
Family
ID: |
23745949 |
Appl.
No.: |
06/439,740 |
Filed: |
November 8, 1982 |
Current U.S.
Class: |
375/240;
341/51 |
Current CPC
Class: |
H03M
5/145 (20130101); H04L 27/3438 (20130101); H03M
13/25 (20130101) |
Current International
Class: |
H04L
27/34 (20060101); H03M 5/14 (20060101); H03M
13/00 (20060101); H03M 13/25 (20060101); H03M
5/00 (20060101); H03K 013/24 () |
Field of
Search: |
;340/347DD ;371/43-45
;375/27,38,39 ;332/9R |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
Ungerboeck, "Channel Coding With Multilevel Phase Signals", IEEE
Transactions & Theory, vol. IT-28, #1, 1/82. .
Forney, "The Viterbi Algorithm", Proceedings of the IEEE, vol. 61,
#3, 3/73..
|
Primary Examiner: Miska; Vit W.
Claims
I claim:
1. Apparatus for sending digital symbols by transmitting discrete
signals corresponding to said symbols over a band-limited channel
using a modulated carrier system, said discrete signals being drawn
from an alphabet of available signals, comprising
a finite-state device that occupies, at any given time, one state
from among a finite number of possible states, said one state
depending on previously sent said symbols, each possible transition
from a previous said state to a current said state being associated
with a subset of said alphabet signals,
signal selection logic comprising means to receive each said symbol
to be sent, to receive a current state from said finite state
device, and to deliver a corresponding said signal selected from
one said subset among a plurality of said subsets whose respective
members are such that at least two of said subsets have at least
one said available signal in common, and at least one said
available signal belongs to only one said subset, and
a modulator for modulating a carrier in accordance with said
delivered signal.
2. The apparatus of claim 1 wherein the number of different
possible said symbols is 2.sup.N, and the size of said signal
alphabet is larger than 2.sup.N and smaller than 2.sup.N+1.
3. The apparatus of claim 1 wherein some said signals (belonging to
a first group) are selected by said signal selection logic without
regard to said current state of said device, and other said signals
(belonging to a second group) are selected by said signal selection
logic based on said current state of said device.
4. The apparatus of claim 1 wherein
said alphabet of available signals comprises a plurality of types
of said signals arranged on a signal plane, the Euclidean distance
in said plane between adjacent said signals belonging to the same
said type being greater than the Euclidean distance in said plane
between adjacent said signals belonging to different said types,
and
said current state depends on the said type to which said selected
signal belongs and on said previous state.
5. The apparatus of claim 4 wherein said available signals are
arranged on a rectangular grid, signals from different said types
being alternated along each of the dimensions of said grid.
6. The apparatus of claim 1 wherein
said subsets each comprise quadruplets of available signals located
at the same distance from the origin of the complex signal plane
but separated by 90.degree. intervals about said origin, and
said signal selection logic quadrantally differentially encodes
said symbols.
7. The apparatus of claim 1 wherein a first group of said alphabet
signals, comprising signals belonging to more than one said subset,
is relatively closer in Euclidean distance to the origin on the
complex signal plane than is a second group of said alphabet
signals comprising signals belonging to only one said subset.
8. The apparatus of claim 7 wherein on the average no more than 1/2
of said signals selected belong to said second group.
9. The apparatus of claim 7 wherein the innermost signal belonging
to said second group is no nearer to said origin than the outermost
signal belonging to said first group.
10. The apparatus of claim 7 further comprising a demodulator
having a decoder of the type which delays the decision of which
signal was transmitted until a number of subsequent signals have
been received, said decision being made based on maximum likelihood
sequence estimation, and wherein said number is selected to assure
to a predetermined probability that said subsequent signals will
include at least one signal belonging to said second group.
11. The apparatus of claim 1 further comprising a demodulator
having a decoder arranged to make tentative decisions about which
signal was transmitted, prior to final decoding of said received
signals, and said demodulator comprises adaptive control circuitry
arranged to be responsive to said tentative decisions.
12. The apparatus of claim 1 wherein said signals are arranged in a
rectangular grid on the complex signal plane.
13. The apparatus of claim 1 wherein said signals are arranged in a
hexagonal grid on the complex signal plane.
14. The apparatus of claim 1 wherein there are no more than 1.5
times 2.sup.N said signals.
15. The apparatus of claim 1 wherein said modulated carrier system
is a double side-band system.
16. The apparatus of claim 1 wherein said modulated carrier system
is a single side-band system.
17. The apparatus of claim 2 wherein N is 6, there are 80 said
signals, there are 2 said states, and said signals are arranged on
a rectangular grid in the complex signal plane.
Description
BACKGROUND OF THE INVENTION
This invention relates to transmitting digital symbols over
band-limited channels using a modulated carrier system of the type
including an encoding circuit for encoding said symbols into
discrete signals selected from an available signal alphabet, and in
which dependencies are introduced between successive signals in the
sequence to increase immunity to noise and distortion.
In conventional data transmission systems in which N-bit data
symbols are each represented by a unique signal drawn from an
alphabet of 2.sup.N such signals, each selection of a signal to be
transmitted depends only upon the data symbol represented by that
signal; there is thus a one-to-one correspondence between the set
of 2.sup.N different data symbols and the signal alphabet.
Csajka et al., U.S. Pat. No. 3,877,768, and Ungerboeck, "Channel
Coding with Multilevel/Phase Signals," IEEE Transactions on
Information Theory, Vol. IT-28, No. 1, January, 1982, describe
systems in which each signal selection depends not only upon the
symbol to be represented, but also upon previous signal selections.
The conventional 2.sup.N signal alphabet is doubled in size to
2.sup.N+1. An encoder maps the 2.sup.N different data symbols into
2.sup.N+1 different coded symbols, each represented as N+1 bits.
Finite-state memory in the encoder causes each (N+1)-bit coded
symbol to depend not only on the current N-bit symbol but also on
the previous data sequence which is reflected in the state of the
finite-state memory. The (N+1)-bit coded symbols are then
represented by a sequence of alphabet signals. The alphabet signals
are divided into subsets which are disjoint (i.e., have no signals
in common), each subset corresponding to transitions from one state
to a particular subsequent state of the finite-state memory. The
effect of the coding is to permit only certain sequences of
alphabet signals to be transmitted, and the coded dependency
information carried by every signal is exploited at the receiver
through use of a maximum likelihood sequence estimation decoding
technique (e.g., one based on the Viterbi Algorithm, as described
in Forney, "The Viterbi Algorithm," Proceedings of the IEEE, Vol.
61, No. 3, March 1973, incorporated herein by reference); in such a
technique, instead of decoding each received signal independently
into the alphabet signal most likely to have been sent (i.e., the
alphabet signal closest to the received signal in the sense of
Euclidean distance, if the noise can be regarded as Gaussian),
decoding decisions are delayed for a predetermined number of signal
intervals to permit each decision to be made in a way that results
in a sequence of received signals being decoded into the sequence
of alphabet signals most likely to have been sent (i.e., into the
sequence of received signals closest in the sense of the algebraic
sum of Euclidean distances or vector Euclidean distance).
The so-called coding gain (i.e., increased resistance to Gaussian
noise) achieved by Csajka and Ungerboeck is of course reduced by
the additional power needed to transmit the larger signal alphabet,
subject to maintaining the same minimum distance between signal
points.
SUMMARY OF THE INVENTION
In general, the invention features, in apparatus in which the
encoding circuit includes a finite-state device whose state depends
on previous symbols and the signal alphabet includes a plurality of
subsets each corresponding to transitions from a particular
previous state to a particular current state, that improvement in
which at least two of the subsets contain at least one signal in
common, and at least one signal belongs to only one of the two
subsets. A substantial coding gain is thus achieved with a
relatively small increase in alphabet size, simplifying
implementation.
In preferred embodiments, the size of the alphabet of symbols is
2.sup.N, and the size of the signal alphabet is larger than 2.sup.N
and smaller than 2.sup.N+1 ; the selection of some signals
(belonging to a first group) is made without regard to the state of
the device, and the selection of other signals (belonging to a
second group) depends on the state of the device; the alphabet
comprises a plurality of types of signals (e.g., different types
arranged alternately along the dimensions of a rectangular grid on
the complex signal plane), the distance between adjacent signals
belonging to the same type being greater than the distance between
adjacent signals belonging to different types, and the current
state depends on the type of each signal selected and on the
previous state of the device, whereby the minimum free distance
between contending signal sequences is increased, permitting a
coding gain to be realized; the subsets each comprise quadruplets
of signals located at the same distance from the origin of the
complex signal plane but separated by 90.degree. intervals about
the origin, and the symbols are quadrantally differentially
encoded, whereby the signal alphabet is immunized from transmission
phase hits of integral multiples of 90.degree.; all of the first
group of alphabet signals are as close or closer to the origin on
the complex signal plane than are the second group, and on average
no more than 1/2 of the signals selected belong to the second
group, whereby the additional transmission power required by the
added signals (in excess of 2.sup.N signals) is relatively small;
there is a demodulator comprising a decoder of the type which
delays the decision of which signal was transmitted until a number
of subsequent signals have been received, the decision being made
based on maximum likelihood sequence estimation, and wherein the
number is selected to assure to a predetermined probability that
the subsequent signals will include at least one signal belonging
to the second group, whereby the coding gain is obtained even
though not every signal carries information determinative of the
sequence of states; the decoder is arranged to make tentative
decisions of which signal was transmitted before all of the
subsequent signals have been received, and the demodulator has
adaptive control circuitry arranged to respond to the tentative
decisions, whereby the control circuitry adapts relatively quickly
even though the final decoder decisions are delayed; the signals
are arranged in a hexagonal grid on the complex signal plane,
whereby a further coding gain may be attained; there are no more
than 1.5 times 2.sup.N signals; the modulated carrier system is
double side-band or single side-band; and N is 6, there are 80
signals, and there are 2 states.
Other advantages and features of the invention will be apparent
from the following description of the preferred embodiment and from
the claims.
DESCRIPTION OF THE PREFERRED EMBODIMENT
We first briefly describe the drawings.
Drawings
FIG. 1 is a block diagram of a transmitter according to the
invention.
FIG. 2 is a block diagram of a transmitter for six-bit data symbols
according to the invention.
FIG. 3 is a rectangular signal point structure for six-bit data
symbols.
FIG. 4 is a state diagram (trellis) for use with the point
structures of FIGS. 3, 6A-6D and 8.
FIG. 5 is a block diagram of a transmitter for six-bit data symbols
including a differential encoder.
FIGS. 6A, 6B, 6C and 6D are alternative signal point structures for
four-bit, five-bit, seven-bit and eight-bit data symbols,
respectively.
FIG. 7A is a hexagonal signal point structure for six-bit
symbols.
FIG. 7B is a trellis for use with FIG. 7A.
FIG. 8 is a one dimensional signal point structure for three-bit
symbols.
STRUCTURE AND OPERATION
Referring to FIG. 1, in general in a transmitter 10 according to
the invention, an incoming sequence of data bits 14 is grouped into
N-bit data symbols 16 using a conventional serial-to-parallel
converter 12. (More generally, the data symbols may be taken from
any discrete alphabet, not necessarily of a size equal to a power
of two and the incoming sequence of bits or of data symbols may
have also undergone some preliminary transformations, such as
scrambling for randomization or security, which are of no
consequence for the present invention.)
The data symbols 16 are encoded in an encoder comprising signal
selection logic 26 and finite-state machine 22 to generate signal
points 28 (which are then conventionally modulated for transmission
over the channel by modulator 32). The signal points 28 depend not
only on the current data symbols 16 but also on the sequence of
previous data symbols. The encoder's memory is the finite-state
machine 22, which at any time is in one of a finite number (M) of
states depending on the history of data symbols, and signal
selection logic 26 (implemented by conventional logic techniques)
determines both the current signal point 28 and the next state 30
as functions of the current data symbol 16 and the current state
20. Every sequence of data symbols thus determines a corresponding
unique sequence of signal points, but not all sequences of signal
points can occur.
The signal alphabet comprises a number of subsets, each subset
corresponding to transitions from one state to a next state of the
finite-state machine, and having its members selected so that the
subsets are at least partially overlapping and at least partially
disjoint; that is, at least two of the subsets contain at least one
signal in common, and at least one signal belongs to only one of
the two subsets.
Because the alphabet subsets overlap, fewer signal points are used
and implementation is simplified. The signal points which are
common to more than one subset are used more often than the other
signal points, so that if the more frequent signal points are
chosen from the signal alphabet as points requiring less power to
transmit, the average power required is less than it would be if
all signal points were equiprobable.
Referring to FIG. 2, in preferred embodiments, with a transmitter
for sending six-bit data symbols, the size of the data symbol
alphabet is 2.sup.N =2.sup.6 =64; and the signal alphabet,
illustrated in FIG. 3, has 80 signal points (i.e., 1.25 times
2.sup.N). Using a simple encoder with only two states, a coding
gain of over 2 dB is obtained, which is better than any previously
known two-state encoding method. Signal selection logic 26 maps
each data symbol 16 into one of the 80 different two-dimensional
signal points 28. The next state (0 or 1) of two-state machine 36
is determined by one bit of information 30 (the two values of which
can be designated S and R) associated with each signal indicating
whether the state should be reversed (R) or remain the same (S)
after that signal is transmitted. The transmitter is clocked at the
modulation rate (or "baud rate") of modulator 32 so that 6 bits per
baud can be transmitted. With quadrature amplitude modulation and
efficient use of bandwidth in modulator 32, the transmitter can
send nominally 6 bits per Hz of channel bandwidth.
The signal selection logic 26 therefore maps the 64 different data
symbols into 80 different signal points using the state of the
two-state machine to determine, as to some of the symbols, which of
two possible signal points is selected.
Referring to FIG. 3, with the 80 signal points 28 arranged on
rectangular signal point grid 50 on the complex signal plane, each
point is identified by a six-bit number (the value of the data
symbol which may be encoded as that point), a letter (R or S), and
one or two numbers in parentheses (indicating the current state or
states of the two-state machine which must exist for the indicated
data symbol to be encoded as that point). The R or S designation
indicates whether the state of the machine is reversed (R) or stays
the same (S) when that signal is selected.
The 80 signal points fall into four subsets of 32 points each,
which can be called S.sub.o, R.sub.o, S.sub.l and R.sub.l. Subset
S.sub.o contains the S-type points which can be reached when the
two-state machine is in state 0. S.sub.l, R.sub.o, and R.sub.l are
defined analogously. S.sub.o and S.sub.l have in common the 24
S-type points which are labeled S(0,1) and fall inside boundary 52,
and differ in their respective eight S-type points which fall
outside boundary 52. R.sub.o and R.sub.l analogously have 24 common
points and each have 8 unique points.
Referring to FIG. 4, trellis 80 (a state transition diagram) shows
the possible states 82 occupied by the two-state machine at each
time in a sequence of discrete times marking the appearances of
successive symbols in the stream. Each vertical pair of circles
represents the two possible states at each time. (The top circle
represents the 0 state and the bottom circle the 1 state.) Time
progresses from left to right.
Trellis 80 also illustrates the possible transitions between
states. A branch 84 leads from each state to each of the two states
which could occur next. Each branch 84 connecting a 0 state to a 0
state is marked S.sub.o to indicate that it represents the subset
of 32 S.sub.o points corresponding to a transition of the two-state
machine from a 0 state to the same 0 state. Analogous meanings
apply to branches R.sub.o, S.sub.l and R.sub.l. Over time the
sequence of transitions (branches) traces a path through the
trellis according to the sequence of states occupied. (Trellis
diagrams are more fully described in the Forney article cited
above.)
Given a starting state, any sequence of data symbols corresponds to
a unique path through the trellis and defines the signals sent and
therefore the states occupied by the two-state machine over
time.
Referring again to FIG. 3, a first group of 48 inner points
(enclosed within boundary 52) are associated respectively with 48
of the 64 possible different data symbols. The data symbol
associated with each inner point is always encoded as that point,
regardless of the state of two-state machine 36 (as indicated by
the appearance of both a 0 and a 1 in the parentheses next to each
inner point).
In a second group comprising the remaining 32 outer points (which
are outside of boundary 52 and thus at least as far from the origin
as any of the inner points), two such outer points are associated
with each of the remaining 16 data symbols. The outer point into
which any of those 16 data symbols is mapped is determined by the
current state of the two-state machine (in accordance with the 0 or
1 which appears in parentheses next to that point). For example,
points 54, 56 are both associated with data symbol 001000; when the
two-state machine is in state 0, data symbol 001000 is encoded as
point 56, otherwise it is encoded as point 54. Thus each
transmitted outer point can be sent only when the two-state machine
is in a particular state. The innermost outer point is as far or
farther from the origin than the outermost inner point.
Throughout the signal structure, the R-type and S-type points are
alternated in the rows and columns of the grid (as shown) so that
each R point has S points as its four nearest neighbors along the
lines of the grid and has R points as its four nearest diagonal
neighbors, and vice versa. Each pair of diagonally adjacent S
points (and each such pair of R points) is separated by a squared
distance of 2d.sub.o.sup.2 (represented for example by line 58 on
FIG. 3), where d.sub.o.sup.2 is the squared distance between two
adjacent points along the grid lines (represented by line 60 on
FIG. 3).
The probability of a random selected data symbol being encoded as
an outer point is p=1/4 (because 16 of the 64 unique data symbols
are associated with outer points), so that on average 1/4 of the
points transmitted will carry information about the state of the
transmitter's two-state machine. The state of the two-state machine
at a given time is equally likely to be 0 or 1 (note that half of
the 80 signal points are S points and half are R points).
Subsets R.sub.o and S.sub.o together are the set of 64 signal
points which can be encoded from state 0, and subsets R.sub.l and
S.sub.l similarly are the set of 64 signal points which can be sent
from state 1. These two sets overlap in the 48 inner points but are
disjoint as to the 32 outer points.
The points are arranged on the grid with 90.degree. quadrantal
symmetry so that, upon rotation of the pattern by any multiple of
90.degree., each S.sub.o point will lie in the same location that
another S.sub.o point occupied before rotation (the S.sub.o subset
thus comprising quadruplets of points), and likewise for S.sub.l,
R.sub.o and R.sub.l points. Because of the symmetry, the receiver
can only derive a reference carrier (for coherent demodulation)
that has the correct phase modulo 90.degree. (i.e., there is a
90.degree. phase ambiguity); therefore, the correct phase may
differ from the received carrier phase by a multiple of 90.degree.
without the receiver being able to detect a discrepancy. This
desirable arrangement (which requires omitting the grid center
point) in combination with quadrantal differential coding
techniques of the kind described in Forney et al., U.S. Pat. No.
3,887,768, incorporated herein by reference, allows correct
decoding despite the 90.degree. phase ambiguity.
Referring to FIG. 5, a differential encoding transmitter has Gray
encoder 70 (which exclusive ORs the most significant two bits of
each data symbol 33), modulo 4 adder 72 and delay 74, the output of
the adder becoming the two most significant bits of the data symbol
used by the coding circuitry 18.
Referring again to FIG. 2, a demodulator 104 includes a decoder 106
in which decoding generally is accomplished by maximum likelihood
sequence estimation (i.e., determining at each point in time which
permissible path through the trellis to the state occupied at that
time is the "closest" to the path actually received and therefore
has the maximum likelihood of being the one which was sent). The
closest path is the one which has the smallest Euclidean distance
from the path received, where the Euclidean distance between a
possible signal path and the received path is the sum of the
Euclidean distances between the respective signal points and
received points on each path.
The decoder may use the Viterbi algorithm (described in Forney
article cited above) to select the maximum likelihood path. In a
preferred embodiment, each received signal is not finally decoded
into a unique data symbol until after a delay of 16 signals
following receipt of the signal in question. The algorithm
determines, as of the current time, the maximum likelihood sequence
to reach each of the two possible current states based on a
contention between the two paths to each state from the two
previous maximum likelihood sequences to the two previous states.
Having determined the two new possible maximum likelihood
sequences, the decoder selects as the 16th previous data symbol the
symbol corresponding to the signal which lies on one or the other
of those sequences at that time. (The number 16 is large enough
that with high probability that signal will be the same for both
contending paths, so that the choice between the two contending
paths is unimportant.) Although the outer points occur only with
frequency 1/4, this rate of occurrence is frequent enough so that
the probability of no outer point appearing within 16 signals is
(3/4).sup.16 or about 1%, which is suitably small. The total number
of signal points, and thus the proportion of outer points, can be
selected to trade off the additional power requirements against the
coding gain produced by the addition of more signal points, and the
delay period can be selected to trade off delay and memory
requirements against higher error probability due to premature
decisions.
The performance of the encoding arrangement in the presence of
Gaussian noise can be determined by principles set forth in the
Forney article, cited above, and is determined primarily by the
free distance, d.sub.free, defined as the minimum squared Euclidean
distance between any two distinct paths through the trellis from
the time they diverge to the time they merge again. With this
coding scheme, the minimum free distance between distinct paths is
2d.sub.o.sup.2. Because the coding scheme assures a minimum free
distance which is twice that of an uncoded system (in which
d.sub.free =d.sub.o.sup.2), the coding scheme can produce a
theoretical coding gain (at low error probabilities) of 3 dB.
This coding gain is obtained because the disjointedness of S.sub.o
and S.sub.l (and R.sub.o and R.sub.l) assures that the average
number of paths that diverge from the sent path at a given time and
are at the free distance from a sent path is finite and is
approximately 16/p=64, where p=1/4 is the probability of an outer
point. The error probability is proportional to this average
number, which is called the error coefficient. Sending an outer
point provides information which aids the receiver in determining
the correct sequence of states of the transmitter's two-state
machine, while sending an inner point provides no such aid.
If only inner points were sent (i.e., if subsets S.sub.o and
S.sub.l (and R.sub.o and R.sub.l) overlapped completely), there
would be infinitely many possible paths through the trellis within
a distance d.sub.free of the sent path (because there would be no
difference between any one path and a second path going through a
complementary sequence of states), and the system would have no
advantage over an uncoded system.
The theoretical coding gain of 3 dB (over an uncoded signal
structure) is reduced by the additional power required by the
additional 16 signal points used in the coded structure (in order
to establish the disjoint subsets, i.e., R.sub.o not equal to
R.sub.l and S.sub.o not equal to S.sub.l). The additional power
required is about 1+p.sup.2 or 1+1/16 or 0.26 dB, leaving a net
asymptotic coding advantage of 2.75 dB over an uncoded system. The
small probability (p) of occurrence of the outer points, while
requiring a relatively small amount of added power, also produces a
higher coefficient (16/p) which further reduces the net coding gain
for non-asymptotic error probabilities. With p=1/2, net coding
gains of 2 dB at error probabilities of the order of 10.sup.-5 are
achieved in the presence of Gaussian noise. For low error
probabilities, p can be kept as small as desired to keep the coding
gain as close to 3 dB as desired. The coding gain is achieved using
only two states in the finite state machine and only 25% more
signal points than in an uncoded system, which enables less complex
and more economical implementation.
Using fewer than 2.sup.N+1 signal points also makes the detected
signal points (before decoding) more reliable, enabling tentative
decoding decisions to be used in the feedback loops of the receiver
(e.g., adaptive equalizer coefficient update, timing recovery, and
carrier recovery) before the final delayed decoding decision is
reached.
FIGS. 6A, 6B, 6C and 6D show rectangular signal structures which
can be used for double side-band (DSB) modulation, with p=1/4, M=2,
and for N=4, 5, 7 and 8, respectively. (In each of FIGS. 6A, 6B, 6C
and 6D, the specific data symbols associated with the signal points
are not identified; the inner points are shown without subscripts
and the outer points are shown with the relevant state subscripts.)
The general rule for arranging the signals on the grid is to use
0.75.times.2.sup.N inner points and 0.5.times.2.sup.N outer points,
breaking them into four subsets S.sub.o, R.sub.o, S.sub.l, R.sub.l
symmetrically.
Referring to FIG. 7A, in other embodiments hexagonal signal
structures may be used. (The closer packing of a hexagonal grid
provides an inherent 0.6 dB advantage over rectangular grids.) FIG.
7A is suitable for N=6, M=3 and DSB modulation. The points are
divided into three types (called A, B and C). Each point of one
type (e.g., an A point) has points of other types (i.e., B and C
points) as its nearest neighbors in the grid and has points of the
same type (i.e., A points) as its next nearest neighbors not along
the grid dimensions. Thus the minimum squared distance between
points of one type is three times the squared distance between
adjacent points on the grid. There are 51 inner points (18 A, 15 B
and 18 C) within boundary 90 and 39 outer points (12 A, 15 B and 12
C). The outer points are arranged in three clusters (respectively
located within boundaries 92, 94, 96), each cluster being
associated with one of the states (0, 1 or 2) of the finite-state
machine, as indicated by a subscript next to each outer point. The
corresponding trellis of FIG. 7B has three possible states in each
symbol interval. In the transmitter, the signal selection circuitry
maps data symbols one-to-one into inner signal points and
one-to-three into outer signal points. The signal selection
circuitry also determines the next state; from any one state, all A
points lead to one next state, all B points to a second next state,
and all C points to a third next state.
The power loss associated with the additional outer points in the
hexagonal structure is approximately 1+2p.sup.2 (slightly higher
than for the rectangular structure) and the coefficient is 18/p,
also slightly higher. The coding gain can approach 3 dB over an
uncoded hexagonal structure, which in turn is inherently about 0.6
dB more efficient than a rectangular grid, being closer packed.
Referring to FIG. 8, in another embodiment a signal structure for
SSB-AM modulation has N=3 and M=2, and the signal points are
arranged along a line with the inner points clustered in the middle
of the line. The corresponding trellis may be the same as the
trellis of FIG. 3.
OTHER EMBODIMENTS
Other embodiments are within the following claims.
For example, the data symbols can be from any discrete alphabet of
any size (not necessarily an integer power of two), the state
machine can be a finite-state machine having any integral number of
states M, the coding structure can be used with
modulation/demodulation schemes other than DSB, the signal point
structure can be arranged hexagonally, rather than rectangularly,
and the number of outer points can be adjusted to change p so that
power loss can be traded for coefficient increase.
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